On the intersection of submonoids of the free monoid

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On the intersection of submonoids of the free
monoid

• L. Giambruno A. Restivo

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Content

• Introduction
• Some useful definitions
• Correspondence submonoids-monoidal automata
• Product of two flower automata
• Prefix case
• Conclusions

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Introduction

• The purpose of our research is to study the
  intersection of two finitely generated submonoids
  of the free monoid on a finite alphabet. In
  particular

• Characterization of the intersection of two free
  submonoids of rank two
• Upper bound for the rank of the intersection of
  two finitely generated submonoids if the
  intersection is finitely generated.

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Motivation

• The study of the intersection of two free
  submonoids of fixed rank is not trivial. In fact
  by a result of M.Latteux and J.Leguy every
  regular language is obtained as omomorphic image
  of the intersection of two finitely generared
  monoids
• 
  Theorem
• Let A be an alphabet and R a
  language of A.
• R is a regular language if
  and only if there exist
• two finite languages F1 , F2
  and a morphism g
• such that
• Karhumakis characterization on the intersection
  of two free submonoids of rank two.
• 
  Theorem (Prefix case)
• Let H,K free submonoids
  of A generated by
• prefix sets of two words.
  If then
• is generated by
  at most two elements.
• If
  then .

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• The Hanna Neumann conjecture for subgroups of
  the free group
• If H,K are finitely
  generated subgroups of a
• free group F then
  ,
• where for every T
  rk(T)max(rk(T)-1, 0)
• with rk(T) the rank of T.
  
• Meakin and Weil in 2002 proved the Hanna Neumann
  conjecture for subgroups positively generated of
  a free group.
• The Hanna Neumann conjecture for submonoids
• Let H,K are finitely
  generated submonoids of A
• with A finite
  alphabet. If is finitely
• generated then
  ,where for
• every T rk(T)max(rk(T)-1,
  0) with rk(T) the rank
• of T.
• We can conjecture also the Hanna Neumann
  conjecture in the free case. In this case by the
  analogies with the conjecture in free groups we
  think that it is more probably true.

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Some useful definition

• The flower automaton
• Let A be a finite alphabet. By Berstel
  and Perrin, given a finite language X in A we
  can associate an automaton AX that recognizes the
  submonoid X of A called the flower automaton.
  

Such automaton has the properties
that 1) It has a unique initial
and final state (i) 2) All
the cycles visit (i) 3) All the
cycles intersect themselves only in (i)
4)The cycles in (i) without (i) as
intermediate vertex have as labels the
words of X.
We can construct AX creating a fixed vertex (i),
creating for every xi in X a cycle in (i) with
label xi and letting (i) be the unique initial
and final vertex.
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Correspondence submonoids- monoidal automata

• Definition Let A(A,Q,i,F,t) be a non
  deterministic trim automaton. A is a monoidal
  automaton if Fi.

Definition Let A(A,Q,i,F,t) be a non
deterministic trim automaton. A is a semiflower
automaton if A is a monoidal automaton such that
every cycle in A visit i.

• If A is a monoidal automaton then HL(A)
  is a submonoid.
• If H is a finitely generated submonoid then
  there exists a monoidal automaton recognizing H.
• Classes closed under this correspondence
• Let A be a monoidal automaton recognizing a
  submonoid H, then
• A is a semiflower automaton if and only if H is a
  finitely generated submonoid
• A is an unambiguos automaton if and only if H is
  a free submonoid
• If A is a deterministic monoidal automaton then H
  is a submonoid generated by a prefix set.

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• Theorem If A is a semi-flower automaton with v
  vertices and e edges and HL(A) then rk(H)
  e-v1. Moreover, if A is unambiguos then rk(H)
  e-v1.
• We remark that a similar result holds for free
  groups.

• Definition Given a graph G, we say that a vertex
  v is a branch point (bp in short) if the number
  of edges going out from v is greater than two.
• In Example 3 the vertex (i) is a bp.

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• Theorem Let A2a,b. If A is a deterministic
  semi-flower automaton with language non empty and
  with v vertices and e edges on A2 then e - v
  bp.
• Remark Trivially if L(A) is empty then e v-1
• Given a submonoid HX with X finite set, let AXD
  be the automaton obtained from AX by applying the
  subset construction and then considering only the
  set of states accessible and coaccessible.

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• Theorem Let HX with X finite prefix set then
  AXD is a
• deterministic semi-flower automaton.

• If X is not a prefix set then AXD is not
  necessarly a semi-flower automaton, as in the
  following example

Given a free finitely generated submonoid T, let
rk(T)max(rk(T)-1, 0). Unifying the previous
results we get Theorem Let HX with X finite
prefix set of A2, then AX D is a deterministic
semi-flower automaton and rk(H) bp. In
Example 4 rk(H)2 bp
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Product of two flower automata

• Let A1 and A2 be two automata and let A1XA2 be
  the product automaton of A1 and A2..
• L(A1 X A2) L(A1) L(A2)
• The product of two automata with a unique initial
  and final state is still an automaton with a
  unique initial and final state
• The product of two deterministic automata is
  still a deterministic automaton
• Remark1 The product of two semi-flower automata
  is not necessarly a semi-flower automaton.
• Remark2 The product of two trim automata is not
  necessarly a trim automaton.
• Given an automaton A, let AT denote the set of
  accessible and coaccessible states of A.

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• Example 6

A2
A1
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• Example 7

A2
A1
b
a
1
2
b
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• We let bp(A1 X A2) be the set of vertices (u,v)
  in A1XA2 bp for A1XA2.
• We let bp(A1 ) X bp(A2) be the set of vertices
  (u,v) in A1XA2 such that u is a bp of A1 and v is
  a bp of A2.
• Lemma 1 If A1 and A2 are deterministic
  automata then
• bp(A1 X A2) bp(A1) X
  bp(A2).
• As remarked before the product of two trim
  automata is not necessarly trim.
• Lemma 2 If A1 and A2 are
  deterministic automata then
• bp(A1 X A2)T
  bp(A1) X bp(A2).

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Prefix case

• Let H and K be submonoids finitely generated by
  prefix sets of A2 .
• All the results proved in the binary alphabet A2
  are also proved in a finite arbitrary alphabet A.
• Let AH and AK be the associated flower automata
  and AHD and AKD as before defined.
• In Example 6 A1 AH with Haab,aba and A2 AK
  Ka,baaba
• In Example 7 A1 AH with Hb,ab and A2
  AK Kb,abbaa,abaa
• Applying Lemma 2 to (AHD X AKD)T we get the
  following theorem

Theorem Let H,K are
submonoids finitely generated by prefix sets of
A2 with A2 a,b. If is finitely
generated then .

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Prefix case rank two

• Theorem
  (Karhumaki,Prefix case)
• Let H,K are free submonoids of A
  generated by prefix
• sets of two words. If
  then is
• generated by at most two
  elements. If
• then .

where aa1a2 , ba2b1
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• Example 4 Xaab,aba , Ya,baaba

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Conclusions

• Not prefix case
• We use the same techniques
• Non-deterministic approach
• Partial result

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Thanks!